Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. ∠ P O R = 2 ∠ P Q R x ∘ = 2 × 50 ∘ x ∘ = 100 ∘

Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. ∠ M N L = 1 2 Reflex ∠ M O L = 1 2 × 260 ∘ x ∘ = 130 ∘

Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. X Y is the diameter of the circle. Therefore ∠ X Z Y = 90 ∘ (Angle on a semi - circle) In ∆ X Y Z x ∘ + 63 ∘ + 90 ∘ = 180 ∘ x ∘ = 27 ∘

Find the value of x ∘ in the following figures:

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of a circle. O A = O B = O C (Radii) In ∆ O A C ∠ O A C = ∠ O C A = 20 ∘ In ∆ O B C , ∠ O B C = ∠ O C B = 35 ∘ (angles opposite to equal sides are equal) ∠ A C B = ∠ O C A + ∠ O C B x ∘ = 20 ∘ + 35 ∘ x ∘ = 55 ∘

If O is the centre of the circle and ∠ A B C = 30 ∘ then find ∠ A O C .

Given ∠ A B C = 30 ∘ ∠ A O C = 2 ∠ A B C (The angle subtended by an arc at the centre is double the angle at any point on the circle) = 2 × 30 ∘ = 60 ∘ Now we shall see, another interesting theorem. We have learnt that minor arc subtends obtuse angle, major arc subtends acute angle and semi circle subtends right angle on the circumference. If a chord A B is given and C and D are two different points on the circumference of the circle, then find ∠ A C B and ∠ A D B . Is there any difference in these angles?

If P Q R S is a cyclic quadrilateral in which ∠ P S R = 70 ∘ and ∠ Q P R = 40 ∘ , then find ∠ P R Q (see Fig. 4.81).

P Q R S is a cyclic quadrilateral Given ∠ P S R = 70 ∘ ∠ P S R + ∠ P Q R = 180 ∘ (the opposite angles of a cyclic quadrilateral are supplementary.) 70 ∘ + ∠ P Q R = 180 ∘ ∠ P Q R = 180 ∘ - 70 ∘ ∠ P Q R = 110 ∘ In ∆ P Q R we have, ∠ P Q R + ∠ P R Q + ∠ Q P R = 180 ∘ (Angle sum property of triangle) 110 ∘ + ∠ P R Q + 40 ∘ = 180 ∘ ∠ P R Q = 180 ∘ - 150 ∘ ∠ P R Q = 30 ∘

In the figure given, find the value of x ∘ and y ∘ .

∠ B A D + ∠ E A D = 180 ° (linear pair) ∠ B A D + 100 ° = 180 ° ∠ B A D = 80 ° Now, ∠ B A D + y ° = 180 ° (Opposite angles of a cyclic quadrilateral is supplementary) y ° = 180 ° - 80 ° = 100 ° In △ B D C 30 ° + y ° + ∠ B D C = 180 ° ∠ B D C = 180 ° - 30 ° + 100 ° = 50 ° Now, ∠ C D F + ∠ C D B + ∠ A D B = 180 ° (Linear pair) 60 ° + 50 ° + ∠ A D B = 180 ° ∠ A D B = 180 ° - 110 ° = 70 ° Again in △ A B D x ° + ∠ A D B + ∠ B A D = 180 ° x ° = 180 ° - 70 ° + 80 ° = 30 °